Another, older, far better known, and permeating almost (if not) all of large scale applications of Computer Science, inadvertent application of coarse geometry is sampling. Indeed, by choosing certain points in a space, and discarding the “less important” points in their neighborhood, in conjunction (in many instances), with the related process of clustering, one in effect discards the small scale structure, as encoded in the discarded vicinities, and rather concentrates on the large scale, i.e. coarse structure. Clearly sampling the “heavy” curvature nodes or edges in a network, and using them as clustering centers is natural. In fact, such clustering based on the combinatorial version of Gaussian curvature, that is the clustering coefficient has been employed for clustering in networks for quite some time now [1]}, and it was only natural to extend this method to a proper notion of metric curvature that allows of the incorporation of weights, be they node or edge weights, accounting for the simultaneous presence of the two types of weights as well [2]}.
Thus we connect the two interpretations of the notion of coarse geometry, namely the global one, motivated largely by Gromov's hyperbolicity, and the local one, (as opposed to the infinitesimal one), obtained by the approximation of manifolds by networks.
We shall show, moreover, that sampling by Ricci curvature, is not just an heuristic, empirical procedure, and it has, in fact, deep mathematical justifications that also point to its full potential. We shall prove our assertion in Section 5 below.
